Saturday, May 17, 2014

There's a minor setback. The fiberglass Dewar has a leak. On the theory side of things, the relativistic range equation is shown to be proper x velocity times the rapidity of the y component of velocity divided by the acceleration due to gravity which is about what you'd expect it to look like if you first looked at the classical result and then squinted. Work is being done to determine what, if anything, to make of the result. Notes and a brief Mathematica file are included.

There is a leak at the Teflon joint that is away from the
Dewar. It’s a rather small leak that can’t
be detected using the roughing pump gauge.

I've very slightly opened the Dewar valve and begun pumping
on the large volume. After about five
minutes, the vacuum was back down to 3 * 10^-1.

It appears that the fiberglass Dewar seals did not reseat. I
opened the valve a little more which I expected to improve the vacuum pressure
if we didn't have a leak since it would increase the available pumping
speed. The pressure got worse and stayed
worse. The mechanical pump is not pulling below
9*10^-1 after about an hour of pumping.

I was wrong yesterday.
The factors to the left of the log are constants, but it’s interesting
to vary beta. When it goes to 1, the
expression goes to infinity. There’s one
other interesting issue though. The
range expression can also be written as:

Written this way, it’s the horizontal component of proper
velocity times the vertical component of rapidity dived by the acceleration of
gravity. The product of the speed of light times the veritical component of
rapidity divided by g gives a proper time that’s equal to the hang time of the projectile. There’s a second expression for the vertical
hang time agreed upon by both MacColl and Shahin. The expression is

converting this to proper time, we get

There’s probably a mistake here somewhere, but I’m not
seeing it today.

Shahin and MacColl’s range equations were compared
[mathematica file] . They’re quite different
notationally, but they are mathematically equivalent.